Geiger's Method | Physics Library

Geiger’s method [1] is an iterative procedure using Gauss-Newton optimization to determine the location of an earthquake, or seismic event. Originally his method was developed to obtain the origin time and Epicentre but it is easily extended to include the Focal Depth for Hypocentre determination.

Given a set of M arrival times t_i find the origin time t₀ and the hypocentre in cartesian coordinatios (x₀,y₀,z₀) which minimize the objective function

M∑ F (X ) = r2i. i=1

(1)

Here, r_i is the difference between observed and calculated arrival times

ri = ti − t0 − Ti,

(2)

and the unknown parameter vector is

X = (t0,x0,y0,z0)T

(3)

In matrix form (1) becomes

F(X ) = rTr

(4)

The Gauss–Newton procedure requires an initial guess of the sought parameters, denoted here as

X ∗ = (t∗0,x∗0,y∗0,z∗0)T,

(5)

which are then used to calculate the adjustment vector

δX = (δt0,δx0,δy0, δz0)T

(6)

ATA δX = − ATr.

(7)

The Jacobian matrix A is defined as

( ∂r ∕ ∂t ∂r ∕∂x ∂r ∕∂y ∂r ∕∂z ) | 1 0 1 0 1 0 1 0 | | ∂r2∕ ∂t0 ∂r2 ∕∂x0 ∂r2∕∂y0 ∂r2 ∕∂z0 | A = |( ... ... ... ... |) . ∂rM ∕∂t0 ∂rM ∕∂x0 ∂rM ∕∂y0 ∂rM ∕∂z0

(8)

The partial derivatives are evaluated at the initial guess, or trial vector, X^∗. Equation (7) can be rewritten as

G δX = g.

(9)

Using (9) and an initial guess X^∗ an adjustment vector can be calculated. The initial guess can then be updated X^∗ + δX and used as the inital guess in the next run of the algorithm. In this manner the sought parameters X can be determined to some tolerance.

References

[1] Geiger, L., “Probability method for the determination of earthquake epicenters from the arrival time only.” Bull. St. Louis Univ. vol. 8, pp. 60-71.

[2] Lee, W. H. K. and Stewart, S. W. Principles and Applications of Microearthquake Networks, Academic Press, New York. 1981

[3] Gibowicz, S. J. and Kijko, A. An Introduction to Mining Seismology, Academic Press, New York. 1994.

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